synopses for massive data: samples, histograms, wavelets

Synopses for Massive Data:

Samples, Histograms,

Wavelets, Sketches

Full text available at: http://dx.doi.org/10.1561/1900000004

Synopses for Massive Data:

Samples, Histograms,

Wavelets, Sketches

Graham Cormode

AT&T Labs — ResearchUSA

[email protected]

Minos Garofalakis

Technical University of CreteGreece

[email protected]

Peter J. Haas

IBM Almaden Research CenterUSA

[email protected]

Chris Jermaine

Rice UniversityUSA

[email protected]

Boston – Delft


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Sketches, Foundation and Trends R© in Databases, vol 4, nos 1–3, pp 1–294, 2011

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Volume 4 Issues 1–3, 2011

Editorial Board

Editor-in-Chief:Joseph M. HellersteinComputer Science DivisionUniversity of California, BerkeleyBerkeley, [email protected]

EditorsAnastasia Ailamaki (EPFL)Michael Carey (UC Irvine)Surajit Chaudhuri (Microsoft Research)Ronald Fagin (IBM Research)Minos Garofalakis (Yahoo! Research)Johannes Gehrke (Cornell University)Alon Halevy (Google)Jeffrey Naughton (University of Wisconsin)Christopher Olston (Yahoo! Research)Jignesh Patel (University of Michigan)Raghu Ramakrishnan (Yahoo! Research)Gerhard Weikum (Max-Planck Institute)


Editorial Scope

Foundations and Trends R© in Databases covers a breadth of top-ics relating to the management of large volumes of data. The journaltargets the full scope of issues in data management, from theoreticalfoundations, to languages and modeling, to algorithms, system archi-tecture, and applications. The list of topics below illustrates some ofthe intended coverage, though it is by no means exhaustive:

• Data Models and QueryLanguages

• Query Processing andOptimization

• Storage, Access Methods, andIndexing

• Transaction Management,Concurrency Control and Recovery

• Deductive Databases

• Parallel and Distributed DatabaseSystems

• Database Design and Tuning

• Metadata Management

• Object Management

• Trigger Processing and ActiveDatabases

• Data Mining and OLAP

• Approximate and InteractiveQuery Processing

• Data Warehousing

• Adaptive Query Processing

• Data Stream Management

• Search and Query Integration

• XML and Semi-Structured Data

• Web Services and Middleware

• Data Integration and Exchange

• Private and Secure DataManagement

• Peer-to-Peer, Sensornet andMobile Data Management

• Scientific and Spatial DataManagement

• Data Brokering andPublish/Subscribe

• Data Cleaning and InformationExtraction

• Probabilistic Data Management

Information for LibrariansFoundations and Trends R© in Databases, 2011, Volume 4, 4 issues. ISSN paperversion 1931-7883. ISSN online version 1931-7891. Also available as a com-bined paper and online subscription.



Vol. 4, Nos. 1–3 (2011) 1–294c© 2012 G. Cormode, M. Garofalakis, P. J. Haas

and C. Jermaine

DOI: 10.1561/1900000004

Synopses for Massive Data: Samples,Histograms, Wavelets, Sketches

Graham Cormode1, Minos Garofalakis2,

Peter J. Haas3, and Chris Jermaine4

1 AT&T Labs — Research, 180 Park Avenue, Florham Park, NJ 07932,USA, [email protected]

2 Technical University of Crete, University Campus — Kounoupidiana,Chania, 73100, Greece, [email protected]

3 IBM Almaden Research Center, 650 Harry Road, San Jose, CA95120-6099, USA, [email protected]

4 Rice University, 6100 Main Street, Houston, TX 77005, USA,[email protected]

Abstract

Methods for Approximate Query Processing (AQP) are essential fordealing with massive data. They are often the only means of provid-ing interactive response times when exploring massive datasets, and arealso needed to handle high speed data streams. These methods proceedby computing a lossy, compact synopsis of the data, and then execut-ing the query of interest against the synopsis rather than the entiredataset. We describe basic principles and recent developments in AQP.We focus on four key synopses: random samples, histograms, wavelets,and sketches. We consider issues such as accuracy, space and time effi-ciency, optimality, practicality, range of applicability, error bounds onquery answers, and incremental maintenance. We also discuss the trade-offs between the different synopsis types.


Contents

1 Introduction 1

1.1 The Need for Synopses 21.2 Survey Overview 41.3 Outline 5

2 Sampling 11

2.1 Introduction 112.2 Some Simple Examples 122.3 Advantages and Drawbacks of Sampling 172.4 Mathematical Essentials of Sampling 212.5 Different Flavors of Sampling 322.6 Sampling and Database Queries 382.7 Obtaining Samples from Databases 562.8 Online Aggregation Via Sampling 65

3 Histograms 69

3.1 Introduction 713.2 One-Dimensional Histograms: Overview 793.3 Estimation Schemes 853.4 Bucketing Schemes 953.5 Multi-Dimensional Histograms 1123.6 Approximate Processing of General Queries 1273.7 Additional Topics 135

ix


4 Wavelets 145

4.1 Introduction 1454.2 One-Dimensional Wavelets and

Wavelet Synopses: Overview 1464.3 Wavelet Synopses for Non-L2 Error Metrics 1564.4 Multi-Dimensional Wavelets 1724.5 Approximate Processing of General Queries 1824.6 Additional Topics 186

5 Sketches 203

5.1 Introduction 2035.2 Notation and Terminology 2055.3 Frequency Based Sketches 2125.4 Sketches for Distinct Value Queries 2425.5 Other Topics in Sketching 259

6 Conclusions and Future Research Directions 265

6.1 Comparison Across Different Methods 2656.2 Approximate Query Processing in Systems 2696.3 Challenges and Future Directions for Synopses 271

Acknowledgments 277

References 279


1

Introduction

A synopsis of a massive dataset captures vital properties of the originaldata while typically occupying much less space. For example, supposethat our data consists of a large numeric time series. A simple summaryallows us to compute the statistical variance of this series: we maintainthe sum of all the values, the sum of the squares of the values, andthe number of observations. Then the average is given by the ratio ofthe sum to the count, and the variance is ratio of the sum of squaresto the count, less the square of the average. An important property ofthis synopsis is that we can build it efficiently. Indeed, we can find thethree summary values in a single pass through the data.

However, we may need to know more about the data than merely itsvariance: how many different values have been seen? How many timeshas the series exceeded a given threshold? What was the behavior ina given time period? To answer such queries, our three-value summarydoes not suffice, and synopses appropriate to each type of query areneeded. In general, these synopses will not be as simple or easy to com-pute as the synopsis for variance. Indeed, for many of these questions,there is no synopsis that can provide the exact answer, as is the casefor variance. The reason is that for some classes of queries, the query

1


2 Introduction

answers collectively describe the data in full, and so any synopsis wouldeffectively have to store the entire dataset.

To overcome this problem, we must relax our requirements. In manycases, the key objective is not obtaining the exact answer to a query,but rather receiving an accurate estimate of the answer. For example,in many settings, receiving an answer that is within 0.1% of the trueresult is adequate for our needs; it might suffice to know that the trueanswer is roughly $5 million without knowing that the exact answer is$5,001,482.76. Thus we can tolerate approximation, and there are manysynopses that provide approximate answers. This small relaxation canmake a big difference. Although for some queries it is impossible toprovide a small synopsis that provides exact answers, there are manysynopses that provide a very accurate approximation for these querieswhile using very little space.

1.1 The Need for Synopses

The use of synopses is essential for managing the massive data thatarises in modern information management scenarios. When handlinglarge datasets, from gigabytes to petabytes in size, it is often impracticalto operate on them in full. Instead, it is much more convenient to build asynopsis, and then use this synopsis to analyze the data. This approachcaptures a variety of use-cases:

• A search engine collects logs of every search made, amount-ing to billions of queries every day. It would be too slow, andenergy-intensive, to look for trends and patterns on the fulldata. Instead, it is preferable to use a synopsis that is guar-anteed to preserve most of the as-yet undiscovered patternsin the data.• A team of analysts for a retail chain would like to study the

impact of different promotions and pricing strategies on salesof different items. It is not cost-effective to give each analystthe resources needed to study the national sales data in full,but by working with synopses of the data, each analyst canperform their explorations on their own laptops.


1.1 The Need for Synopses 3

• A large cellphone provider wants to track the health ofits network by studying statistics of calls made in differentregions, on hardware from different manufacturers, under dif-ferent levels of contention, and so on. The volume of infor-mation is too large to retain in a database, but instead theprovider can build a synopsis of the data as it is observedlive, and then use the synopsis off-line for further analysis.

These examples expose a variety of settings. The full data may residein a traditional data warehouse, where it is indexed and accessible, butis too costly to work on in full. In other cases, the data is stored as flat-files in a distributed file system; or it may never be stored in full, butbe accessible only as it is observed in a streaming fashion. Sometimessynopsis construction is a one-time process, and sometimes we needto update the synopsis as the base data is modified or as accuracyrequirements change. In all cases though, being able to construct a highquality synopsis enables much faster and more scalable data analysis.

From the 1990s through today, there has been an increasing demandfor systems to query more and more data at ever faster speeds. Enter-prise data requirements have been estimated [173] to grow at 60% peryear through at least 2011, reaching 1,800 exabytes. On the other hand,users — weaned on Internet browsers, sophisticated analytics and sim-ulation software with advanced GUIs, and computer games — havecome to expect real-time or near-real-time answers to their queries.Indeed, it has been increasingly realized that extracting knowledgefrom data is usually an interactive process, with a user issuing a query,seeing the result, and using the result to formulate the next query,in an iterative fashion. Of course, parallel processing techniques canalso help address these problems, but may not suffice on their own.Many queries, for example, are not embarrassingly parallel. More-over, methods based purely on parallelism can be expensive. Indeed,under evolving models for cloud computing, specifically “platform as aservice” fee models, users will pay costs that directly reflect the com-puting resources that they use. In this setting, use of ApproximateQuery Processing (AQP) techniques can lead to significant cost savings.Similarly, recent work [15] has pointed out that approximate processing


4 Introduction

techniques can lead to energy savings and greener computing. ThusAQP techniques are essential for providing, in a cost-effective manner,interactive response times for exploratory queries over massive data.

Exacerbating the pressures on data management systems is theincreasing need to query streaming data, such as real time financialdata or sensor feeds. Here the flood of high speed data can easily over-whelm the often limited CPU and memory capacities of a stream pro-cessor unless AQP methods are used. Moreover, for purposes of networkmonitoring and many other applications, approximate answers sufficewhen trying to detect general patterns in the data, such as a denial-of-service attack. AQP techniques are thus well suited to streaming andnetwork applications.

1.2 Survey Overview

In this survey, we describe basic principles and recent developments inbuilding approximate synopses (i.e., lossy, compressed representations)of massive data. Such synopses enable AQP, in which the user’s queryis executed against the synopsis instead of the original data. We focuson the four main families of synopses: random samples, histograms,wavelets, and sketches.

A random sample comprises a “representative” subset of the datavalues of interest, obtained via a stochastic mechanism. Samples canbe quick to obtain, and can be used to approximately answer a widerange of queries.

A histogram summarizes a dataset by grouping the data values intosubsets, or “buckets,” and then, for each bucket, computing a small setof summary statistics that can be used to approximately reconstruct thedata in the bucket. Histograms have been extensively studied and havebeen incorporated into the query optimizers of virtually all commercialrelational DBMSs.

Wavelet-based synopses were originally developed in the contextof image and signal processing. The dataset is viewed as a set ofM elements in a vector — that is, as a function defined on the set{0,1,2, . . . ,M − 1} — and the wavelet transform of this function isfound as a weighted sum of wavelet “basis functions.” The weights,


1.3 Outline 5

or coefficients, can then be “thresholded,” for example, by eliminatingcoefficients that are close to zero in magnitude. The remaining smallset of coefficients serves as the synopsis. Wavelets are good at capturingfeatures of the dataset at various scales.

Sketch summaries are particularly well suited to streaming data.Linear sketches, for example, view a numerical dataset as a vector ormatrix, and multiply the data by a fixed matrix. Such sketches are mas-sively parallelizable. They can accommodate streams of transactions inwhich data is both inserted and removed. Sketches have also been usedsuccessfully to estimate the answer to COUNT DISTINCT queries, anotoriously hard problem.

Many questions arise when evaluating or using synopses.

• What is the class of queries that can be approximatelyanswered?• What is the approximation accuracy for a synopsis of a given

size?• What are the space and time requirements for constructing

a synopsis of a given size, as well as the time required toapproximately answer the query?• How should one choose synopsis parameters such as the num-

ber of histogram buckets or the wavelet thresholding value?Is there an optimal, that is, most accurate, synopsis of a givensize?• When using a synopsis to approximately answer a query, is

it possible to obtain error bounds on the approximate queryanswer?• Can the synopsis be incrementally maintained in an efficient

manner?• Which type of synopsis is best for a given problem?

We explore these issues in subsequent chapters.

1.3 Outline

It is possible to read the discussion of each type of synopsis in isolation,to understand a particular summarization approach. We have tried touse common notation and terminology across all chapters, in order to


6 Introduction

facilitate comparison of the different synopses. In more detail, the topicscovered by the different chapters are given below.

1.3.1 Sampling

Random samples are perhaps the most fundamental synopses for AQP,and the most widely implemented. The simplicity of the idea — exe-cuting the desired query against a small representative subset of thedata — belies centuries of research across many fields, with decadesof effort in the database community alone. Many different methods ofextracting and maintaining samples of data have been proposed, alongwith multiple ways to build an estimator for a given query. This chap-ter introduces the mathematical foundations for sampling, in termsof accuracy and precision, and discusses the key sampling schemes:Bernoulli sampling, stratified sampling, and simple random samplingwith and without replacement.

For simple queries, such as basic SUM and AVERAGE queries, it isstraightforward to build unbiased estimators from samples. The moregeneral case — an arbitrary SQL query with nested subqueries — ismore daunting, but can sometimes be solved quite naturally in a pro-cedural way.

For small tables, drawing a sample can be done straightforwardly.For larger relations, which may not fit conveniently in memory, or maynot even be stored on disk in full, more advanced techniques are neededto make the sampling process scalable. For disk-resident data, sam-pling methods that operate at the granularity of a block rather than atuple may be preferred. Existing indices can also be leveraged to helpthe sampling. For large streams of data, considerable effort has beenput into maintaining a uniform sample as new items arrive or exist-ing items are deleted. Finally, “online aggregation” algorithms enhanceinteractive exploration of massive datasets by exploiting the fact thatan imprecise sampling-based estimate of a query result can be incre-mentally improved simply by collecting more samples.

1.3.2 Histograms

The histogram is a fundamental object for summarizing the frequencydistribution of an attribute or combination of attributes. The most


1.3 Outline 7

basic histograms are based on a fixed division of the domain(equi-width), or using quantiles (equi-depth), and simply keep statisticson the number of items from the input which fall in each such bucket.But many more complex methods have been designed, which aim toprovide the most accurate summary possible within a limited spacebudget. Schemes differ in how the buckets are chosen, what statisticsare stored, how estimates are extracted, and what classes of query aresupported. They are quantified based on the space and time require-ments used to build them, and the resulting accuracy guarantees thatthey provide.

The one-dimensional case is at the heart of histogram construction,since higher dimensions are typically handled via extensions of one-dimensional ideas. Beyond equi-width and equi-depth, end biased andhigh biased, maxdiff and other generalizations have been proposed. Fora variety of approximation-error metrics, dynamic programming (DP)methods can be used to find histograms — notably the “v-optimalhistograms” — that minimize the error, subject to an upper bound onthe allowable histogram size. Approximate methods can be used whenthe quadratic cost of DP is not practical. Many other constructions,both optimal and heuristic, are described, such as lattice histograms,STHoles, and maxdiff histograms. The extension of these methodsto higher dimensions adds complexity. Even the two-dimensional casepresents challenges in how to define the space of possible bucketings.The cost of these methods also rises exponentially with the dimen-sionality of the data, inspiring new approaches that combine sets oflow-dimensional histograms with high-level statistical models.

Histograms most naturally answer range–sum queries — for exam-ple, “compute total sales between July and September for adults fromage 25 through 40” — and their variations. They can also be used toapproximate more general classes of queries, such as aggregations overjoins. Various negative theoretical and empirical results indicate thatone should not expect histograms to give accurate answers to arbitraryqueries. Nevertheless, due to their conceptual simplicity, histograms canbe effectively used for a broad variety of estimation tasks, including set-valued queries, real-valued data, and aggregate queries over predicatesmore complex than simple ranges.


8 Introduction

1.3.3 Wavelets

The wavelet synopsis is conceptually close to the histogram summary.The central difference is that, whereas histograms primarily producebuckets that are subsets of the original data-attribute domain, wave-let representations transform the data and seek to represent the mostsignificant features in a wavelet (i.e., “frequency”) domain, and cancapture combinations of high and low frequency information. The mostwidely discussed wavelet transformation is the Haar-wavelet transform(HWT), which can, in general, be constructed in time linear in thesize of the underlying data array. Picking the B largest HWT coeffi-cients results in a synopsis that provides the optimal L2 (sum-squared)error for the reconstructed data. Extending from one-dimensional tomulti-dimensional data, as with histograms, provides more definitionalchallenges. There are multiple plausible choices here, as well as algo-rithmic challenges in efficiently building the wavelet decomposition.

The core AQP task for wavelet summaries is to estimate the answerto range sums. More general SPJ (select, project, join) queries can alsobe directly applied on relation summaries, to generate a summary ofthe resulting relation. This is made possible through an appropriately-defined AQP algebra that operates entirely in the domain of waveletcoefficients.

Recent research into wavelet representations has focused on errorguarantees beyond L2. These include L1 (sum of errors) or L∞(maximum error), as well as relative-error versions of these measures.A fundamental choice here is whether to restrict the possible coef-ficient values to those arising under the basic wavelet transform, orto allow other (unrestricted) coefficient values, specifically chosen toreduce the target error metric. The construction of such (restricted orunrestricted) wavelet synopses optimized for non-L2 error metrics is achallenging problem.

1.3.4 Sketches

Sketch techniques have undergone extensive development over thepast few years. They are especially appropriate for streaming data,in which large quantities of data flow by and the sketch summary must


1.3 Outline 9

continually be updated quickly and compactly. Sketches, as presentedhere, are designed so that the update caused by each new piece of datais largely independent of the current state of the summary. This designchoice makes them faster to process, and also easy to parallelize.

“Frequency based sketches” are concerned with summarizing theobserved frequency distribution of a dataset. From these sketches, accu-rate estimations of individual frequencies can be extracted. This leadsto algorithms for finding approximate “heavy hitters” — items thataccount for a large fraction of the frequency mass — and quantilessuch as the median and its generalizations. The same sketches can alsobe used to estimate the sizes of (equi)joins between relations, self-joinsizes, and range queries. Such sketch summaries can be used as prim-itives within more complex mining operations, and to extract waveletand histogram representations of streaming data.

A different style of sketch construction leads to sketches for“distinct-value” queries that count the number of distinct values ina given multiset. As mentioned above, using a sample to estimate theanswer to a COUNT DISTINCT query may give highly inaccurateresults. In contract, sketching methods that make a pass over the entiredataset can provide guaranteed accuracy. Once built, these sketchesestimate not only the cardinality of a given attribute or combinationof attributes, but also the cardinality of various operations performedon them, such as set operations (union and difference), and selectionsbased on arbitrary predicates.

In the final chapter, we compare the different synopsis methods.We also discuss the use of AQP within research systems, and discusschallenges and future directions.

In our discussion, we often use terminology and examples thatarise in classical database systems, such as SQL queries over relationaldatabases. These artifacts partially reflect the original context of theresults that we survey, and provide a convenient vocabulary for the var-ious data and access models that are relevant to AQP. We emphasizethat the techniques discussed here can be applied much more generally.Indeed, one of the key motivations behind this survey is the hope thatthese techniques — and their extensions — will become a fundamentalcomponent of tomorrow’s information management systems.


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